problem set 10, solution.pdf

Problem set 10, solutions

1. Suppose two economic agents i ∈ {1, 2} have utility ui : R → R of theform ui(x) = −(x− bi)2, where bi ∈ R is a parameter. Suppose b1 < b2. Anallocation is simply x ∈ R.

1. Find all Pareto efficient allocations.

2. Solve maxx∈R u1(x) subject to u2(x) ≥ u2 for all values of u2 ∈ R.Show that the set of solutions to the maximization problem, as u2

varies, is the set of Pareto efficient allocations from the first part.

solution: 1. Any x /∈ [b1, b2] is clearly Pareto inefficient. To see this,suppose x /∈ [b1, b2] is Pareto efficient. Suppose x < b1. Then ui(x) < ui(b1)for ∀i ∈ {1, 2}. Suppose x > b2. Then ui(x) < ui(b2) for ∀i ∈ {1, 2}.To see that any x ∈ [b1, b2] is Pareto efficient, fix x ∈ [b1, b2] and considerx′ ∈ [b1, b2] such that x′ 6= x. If x′ < x, then u2(x′) < u2(x) and if x′ > x,then u1(x′) < u1(x). Hence x is Pareto efficient. Because the choice ofx ∈ [b1, b2] was arbitrary, any x ∈ [b1, b2] is Pareto efficient.

solution: 2. For u2 > 0 the constraint u2(x) ≥ u2 can never be satisfied,so consider u2 ≤ 0. The Lagrangian of the problem is

L(x, λ) = −(x− b1)2 − λ[−(−(x− b2)2) + u2]

with the first order and complementary slackness conditions

−2(x∗ − b1)− 2λ∗(x∗ − b2) = 0

λ∗[(x∗ − b2)2 + u2] = 0.

When λ∗ = 0, then x∗ = b1. When λ∗ > 0, then (x∗ − b2)2 = −u2, whichrewrites as x∗ = b2 ±

√−u2. Clearly, only the − applies (if x∗ ≥ b2 then

the first order condition can never be satisfied), so that x∗ = b2 −√−u2.

The constraint does not bind if u2 ≤ −(b1 − b2)2. For u2 ∈ [−(b1 − b2)2, 0],x∗ varies in between x∗ = b2 −

√(b1 − b2)2 = b2 − (b2 − b1) = b1 and

x∗ = b2 −√

0 = b2, so that x∗ ∈ [b1, b2].

2. Suppose two economic agents i ∈ {1, 2} have utility ui : R → R ofthe form ui(x, y) = −(x − bxi)

2 − (y − byi)2, where bxi ∈ R and byi ∈ R

are parameters. Suppose bx1 < bx2 and by1 < by2. An allocation is pair(x, y) ∈ R2.

1. Show that if (x∗, y∗) is a Pareto efficient allocation, then x∗ ∈ [bx1, bx2]and y∗ ∈ [by1, by2].

2. Solve max(x,y)∈R2 u1(x, y) subject to u2(x, y) ≥ u2 for all values ofu2 ∈ R. Draw the set of solutions in (x, y) space.

1


solution: 1. The desired results follows by similar argument as in theprevious question. If x /∈ [bx1, bx2] or y /∈ [by1, by2], then (x′, y′) will satisfyu1(x′, y′) > u1(x, y) and u2(x′, y′) > u2(x, y) if we pick x′ = bx1 (if x < bx1)or x′ = bx2 (if x > bx2) or y′ = by1 (if y < by1) or y′ = by2 (if y > by2).

solution: 2. If u2 > 0 then the constraint u2(x, y) ≥ u2 can never besatisfied so consider u2 ≤ 0. The Lagrangian of the problem is

L(x, y, λ) = −(x− bx1)2 − (y − by1)2 − λ[(x− bx2)2 + (y − by2)2 + u2]

with the first order and complementary slackness conditions

−2(x∗ − bx1)− 2λ∗(x∗ − bx2) = 0

−2(y∗ − by1)− 2λ∗(y∗ − by2) = 0

λ∗[(x∗ − bx2)2 + (y∗ − by2)2 + u2] = 0.

If λ∗ = 0, then x∗ = bx1 and y∗ = by1. If λ∗ > 0 then u2 + (x∗ − bx2)2 +(y∗ − by2)2 = 0 and from the first two first order conditions

x− bx2 = (y − by2)bx2 − bx1

by2 − by1.

Substituting this into the constraint, and denoting 1 +(bx2−bx1by2−by1

)2= c > 0,

y∗ = yb2 −√−u2c and hence x∗ = bx2 −

√−u2(c−1)

c .

The constraint does not bind if u2 ≤ −(bx1 − bx2)2 − (by1 − by2)2. Foru2 ∈ [−(bx1 − bx2)2 − (by1 − by2)2, 0], (x∗, y∗) varies between

x∗ = bx2 −

√√√√√√ [(bx1 − bx2)2 + (by1 − by2)2](bx2−bx1by2−by1

)2

1 +(bx2−bx1by2−by1

)2 = bx2 − (bx2 − bx1) = bx1

y∗ = yb2 −√√√√ [(bx1 − bx2)2 + (by1 − by2)2]

1 +(bx2−bx1by2−by1

)2 = by2 − (by2 − by1) = by1

and x∗ = bx2, y∗ = by2. From x− bx2 = (y− by2) bx2−bx1by2−by1 we also know that x

is linear in y so that the set of solutions to the maximization problem drawnin the (x, y) space is a straight line with endpoints (xb1, yb1) and (xb2, yb2).

2


y

xbx1

by1

bx2

by2

3. Suppose the demand for a commodity is c−pdb , where b > 0, c > 0 and

pd is the price consumers pay for the commodity. Suppose the supply of thecommodity is ps

a , where a > 0 and ps is the price producers charge for eachunit of the commodity sold. In addition, the commodity is subject to a tax,whereby each unit of the commodity traded is subject to a tax of t > 0.

1. Suppose the rule is that the producers send t to the government foreach unit of the commodity sold. Calculate the market equilibriumquantity traded and the equilibrium prices.

2. Suppose the rule is that the consumers send t to the government foreach unit of the commodity sold. Calculate the market equilibriumquantity traded and the equilibrium prices.

3. In each of the cases above, calculate the revenue of the tax, the re-duction in consumer surplus stemming from taxation, the reductionin producer surplus stemming from taxation and the deadweight losscaused by taxation.

4. Calculate the share of the revenue paid by the consumers (equilibriumpost-tax quantity times the increase in the pd caused by taxation).Calculate the share of the revenue paid the producers.

5. Calculate the elasticity of the supply, εs, and of the demand, εd, withrespect to price. Show that for small t the shares calculated in theprevious part are equal to εs

εs−εd and −εdεs−εd (that is, proportional to the

relative magnitude of the elasticities).

solution: 1. Suppose the commodity is traded for p. Then the producersreceive p− t per unit sold and the supply is p−t

a . The consumers pay p perunit bought and the demand is c−p

b . The equilibrium price equates supply

and demand and hence is p∗ = ca+tba+b . The consumers pay p∗d = p∗ = ca+tb

a+bper unit bought and the producers net revenue per unit sold is p∗s = p∗− t =ca−taa+b . The equilibrium quantity traded is x∗ = c−t

a+b .

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solution: 2. Suppose the commodity is traded for p. Then the producersreceive p per unit sold and the supply is p

a . The consumers pay p + t

per unit bought and the demand is c−p−tb . The equilibrium price equates

supply and demand and hence is p∗ = ca−taa+b . The consumers pay in total

p∗d = p∗ + t = ca+tba+b per unit bought and the producers net revenue per unit

sold is p∗s = p∗ = ca−taa+b . The equilibrium quantity traded is x∗ = c−t

a+b .Clearly the equilibrium quantity is unaffected by the identity of the agent

who sends the tax to the government. The same is true for the price con-sumers end up paying and for the net price per unit producers receive.What is important is that the price that enters the supply function is differ-ent than the price that enters the demand function. Whether it is writtenas ps + t = pd or ps = pd − t is irrelevant.

solution: 3. The revenue the tax raises is R(t) = t c−ta+b . The consumer

surplus with taxation is(c−p∗d)x∗

2 = b2

(c−t)2(a+b)2

. The reduction in consumer

surplus caused by taxation is thus ∆CS(t) = b2

(c)2

(a+b)2− b

2(c−t)2(a+b)2

= bct− t

2

2(a+b)2

.

The producer surplus with taxation is p∗sx∗

2 = a2

(c−t)2(a+b)2

. The reduction of

producer surplus caused by taxation is thus ∆PS(t) = a2

(c)2

(a+b)2− a

2(c−t)2(a+b)2

=

act− t

2

2(a+b)2

. Note that ∆CS(t) + ∆PS(t) =ct− t

2

2a+b . This gives deadweight loss

of taxation DWL(t) = ∆CS(t) + ∆PS(t)−R(t) = t2

2(a+b) . As expected, thedeadweight loss is linear in the tax squared.

solution: 4. The increase of p∗d caused by taxation is p∗d−caa+b = tb

a+b . The

revenue paid by consumers is thus tba+bx

∗ = tba+b

c−ta+b . The share of revenue

paid by consumers is thustba+b

c−ta+b

t c−ta+b

= ba+b . The decrease of p∗s caused by

taxation is caa+b − p

∗s = ta

a+b . The revenue paid by producers is thus taa+bx

∗ =

taa+b

c−ta+b . The share of revenue paid by producers is thus

taa+b

c−ta+b

t c−ta+b

= aa+b .

solution: 5. Elasticity of the supply function is εs =∂ psa

∂pspspsa

= 1. Elasticity

of the demand function is εd =∂c−pdb

∂pd

pdc−pdb

= − pdc−pd . Substituting into

εsεs−εd gives εs

εs−εd = 11+

pdc−pd

= c−pdc . Substituting p∗d = ca+tb

a+b gives εsεs−εd =

ca+cb−ca−tbc(a+b) = cb−tb

c(a+b) . This becomes ba+b when t = 0. Similarly −εd

εs−εd = ca+tbc(a+b)

becomes aa+b when t = 0.

The intuition behind these results is that the inelastic part of the marketin general bears the larger share of the tax burden and the elastic part ofthe market bears the smaller share of the tax burden. When εs is large,

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then εsεs−εd will be close to unity and the consumers will bear a large share.

When, on the other hand, −εd is large, then εsεs−εd will be close to zero and

the consumers will bear a small share.

4. Consider an economy composed of a single firm, single consumer, andtwo goods m and x. Normalize the price of m to unity. The price of x isp ≥ 0.

The utility of the consumer is u : R × R≥0 → R and if she demands(consumes) md of the m good and xd of the x good, her utility is u(md, xd) =md + φ

√xd, where φ > 0. Her endowment of the x commodity is zero and

her endowment of the m commodity is ωm > 0. She is the sole owner of thefirm and hence her income is ωm · 1 + π, where π is the firm’s profit.

The firm produces good x using good m as an input. Its productionfunction is f : R≥0 → R. If the firm uses ms units of the m good, herproduction of the x good is f(ms) =

√msδ , where δ > 0.

1. Denote by y ≥ 0 the amount of m used in the production of x. Asyou vary y, the remaining amount of m, ωm − y and the amount ofx produced, f(y), trace out the production possibility frontier of theeconomy. Draw the production possibility frontier. Calculate its slope

as a function of x produced, that is as a function of√

yδ .

2. On the production possibility frontier calculate the utility of the con-sumer if she consumes all m and x produced (as a function of y).Maximize this utility with respect to y ≥ 0. The resulting levels ofm and x are the Pareto efficient allocation. Calculate the slope of theconsumer’s indifference curve and show that it is equal to the slope ofthe production possibility frontier at the Pareto efficient allocation.

3. Solve the consumer’s utility maximization problem if she takes themarket conditions as given. Write the amount of x demanded by theconsumer, xd(p, ωm, π) and the amount of m demanded, md(p, ωm, π).

4. Solve the firm’s profit maximization problem. Write the amount of xsupplied, xs(p), and the amount of m used in the production, ms(p).Write down the profit of the firm π(p).

5. Find the market equilibrium price p∗, the resulting consumptions levelsof the consumer, x∗d and m∗d, the resulting production levels, x∗s andm∗s, and the resulting profit π∗. Is the equilibrium Pareto efficient?

6. What is the intuition behind the comparative static effect of δ and φon the variables from the previous part.

5


solution: 1. When y amount of good m is used in the production of the x

good, we are at a point (ωm − y, f(y) =√

yδ ) on the production possibility

frontier. We can think of the production possibility frontier as of a function

taking z = ωm − y ∈ (−∞, ωm] into f(ωm − z) =√

ωm−zδ with derivative

− 12δ

1√ωm−zδ

= − 12δ

√δy = − 1

2δ1xs

, if we denote the amount of x produced by

xs. An example of the production possibility frontier is below.x

m

solution: 2. For y ≥ 0 the utility of the consumer is

ωm − y + φ

√√yδ = ωm − y + φ

y14

δ14

which is maximized at ype =[φ4

28δ

] 13

so that the resulting production of x is

xpe =[φ

22δ

] 23. Hence (xpe, ωm − ype) is the Pareto efficient allocation.

The slope of the consumer’s indifference curve at any (md, xd) is −2√xdφ .

Some algebra shows that

−2√xpe

φ= −

[2

φ2δ

] 13

− 1

2δ

1

xpe= −

[2

φ2δ

] 13

so that the slopes of the consumer’s indifference curve and of the productionpossibility frontier are the same at the Pareto efficient allocation.

solution: 3. The consumer chooses (xd,md) taking p and ωm+π as given.Note that in the market equilibrium π will be a function of p and of the(equilibrium) amount of x supplied, but all this is taken as given at thispoint. The consumer’s problem is

maxxd≥0,md∈R

md + φ√xd s.t. md + pxd ≤ ωm + π

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The two first order conditions are (we are skipping some steps here, namelythose that take into account the xd ≥ 0 constraint, we will see that x∗d willsatisfy x∗d ≥ 0)

1− λ∗ = 0

φ

2√x∗d− λ∗p = 0

so that xd(p, ωm, π) = φ2

4p2. Because λ∗ = 1 the consumer’s budget constraint

has to bind in optimum and hence md(p, ωm, π) = ωm +π− pxd(p, ωm, π) =

ωm + π − φ2

4p .

solution: 4. The firm maximizes its profit pf(ms)−ms taking p as given.Its problem is

maxms∈R≥0

pf(ms)−ms = maxms∈R≥0

p√

msδ −ms

with first order condition 12

√δm∗s

pδ −1 = 0. Solving for m∗s gives ms(p) = p2

4δ .

Substituting ms(p) into the production function gives xs(p) = f(ms(p)) =p2δ . The profit is thus π(p) = pxs(p)−ms(p) = p2

4δ .

solution: 5. There are several ways to find the equilibrium p∗.Approach 1: Substituting π(p) into the consumer’s demand for m gives

ms(p, ωm, π(p)) = ωm+ p2

4δ −φ2

4p . We know the firm uses ms(p) = p2

4δ . Marketm clearing then requires

ωm +(p∗)2

4δ− φ2

4p∗+

(p∗)2

4δ= ωm

which solved for p∗ gives p∗ =[φ2δ2

] 13. We do not need to check market

x clearing since we know that if all but one markets clear (and budgetconstraints bind), the one has to as well.

Approach 2: Directly from xs(p∗) = xd(p

∗, ωm, π) since the right handside of the equation is independent of ωm and π. The equation writes as

p∗

2δ=

φ2

4(p∗)2

and again solved for p∗ gives p∗ =[φ2δ2

] 13. A similar remark about market

clearing as in the previous case applies here as well.

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Substituting p∗ into the expressions above gives

xd(p∗, ωm, π) =

φ2

4(p∗)2=

[φ

22δ

] 23

= x∗d

md(p∗, ω, π(p∗)) = ωm + π(p∗)− φ2

4p∗= ωm −

[φ4

28δ

] 13

= m∗d

xs(p∗) =

p∗

2δ=

[φ

22δ

] 23

= x∗s

ms(p∗) =

(p∗)2

4δ=

[φ4

28δ

] 13

= m∗s

π(p∗) =(p∗)2

4δ=

[φ4

28δ

] 13

= π∗.

It is immediate that all markets clear. It is also immediate that the equilib-rium is Pareto efficient.

solution: 6. When φ increases, the x good receives higher weight in theconsumer’s utility. Hence x∗d increases (despite its price increasing, but thisshould be seen as a result of the higher demand for x). More x has to beproduced so that x∗s increases. This requires more m in the production of xso that m∗s increases. The firm, by providing transformation of m to x forthe consumer is more important for the consumer and hence π∗ increases.In order to satisfy the feasibility constraint, m∗d has to decrease. Note thatincrease in φ can be seen as a (positive) demand shock. Those are typicallyassociated with higher (equilibrium) prices and quantities.

When δ increases, more units of m are required to produce one unitof x. As a result x becomes more expensive and the consumer re-optimizestowards m. As a result x∗d decreases and m∗d increases. Since x∗d decreases x∗shas to as well and subsequently the input in production of x, m∗s. The firmis less important for the consumer and π∗ decreases. Notice that increase inδ can be seen as a (negative) supply shock. Those are typically associatedwith higher equilibrium price but lower quantities.

5. Consider an economy composed of a single firm, two consumers, andtwo goods m and x. Normalize the price of m to unity. The price of x isp ≥ 0.

The utility of consumer i ∈ {1, 2} is ui : R × R≥0 → R and if shedemands (consumes) mdi of the m good and xdi of the x good, her utilityis ui(mdi, xdi) = mdi + φi−1√xdi, where φ > 0. Her endowment of thex commodity is zero and her endowment of the m commodity is ωmi > 0.Consumer i ∈ {1, 2} is a partial owner of the firm and is entitled to θi ∈ [0, 1]share of the firm’s profit. Naturally, θ1 + θ2 = 1.

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The firm produces good x using good m as an input. Its productionfunction is f : R≥0 → R. If the firm uses ms units of the m good, herproduction of the x good is f(ms) =

√msδ , where δ > 0.

1. Find the Pareto efficient allocation(s). [Hint: Maximize, by choos-ing (xd1,md1, xd2,md2, y), u1(md1, xd1) subject to u2(md2, xd2) ≥ u2,ωm1 + ωm2 = md1 +md2 + y, f(y) = xd1 + xd2 and y ≥ 0.]

2. Solve the consumers’ utility maximization problem if they take themarket conditions as given. Write their amount of x and m demanded,xdi(p, ωm, π) and mdi(p, ωm, π).

3. Solve the firm’s profit maximization problem. Write the amount of xsupplied, xs(p), and the amount of m used in the production, ms(p).Write down the profit of the firm π(p).

4. Find the market equilibrium price p∗, the resulting consumptions levelsof the consumers, x∗di and m∗di, the resulting production levels, x∗s andm∗s, and the resulting profit π∗. Is the equilibrium Pareto efficient?

5. What is the intuition behind the comparative static effect of δ and φon the variables from the previous part.

solution: 1. Before following the hint, let me explain why the currentapproach always finds the set of Pareto efficient allocations. Pareto efficiencyis defined as the absence of another allocation that makes all agents as wellof and at least one of them strictly better off. The maximization problemhinted involves maximization of utility of consumer 1 subject to not hurtingconsumer 2. Hence its solution has to be Pareto efficient. By varying u2 onefinds all Pareto efficient allocations.

The maximization problem formally writes as

maxxd1 ≥ 0, xd2 ≥ 0, y ≥ 0md1 ∈ R,md2 ∈ R

md1 +√xd1

s.t. md2 + φ√xd2 ≥ u2

s.t. ωm1 + ωmd2 = y +md1 +md2

s.t. f(y) =√

yδ = xd1 + xd2

and hence the Lagrangian is

L(xd1, xd2,md1,md2, y, λ) = md1 +√xd1

− λ[−md2 − φ√xd2 + u2]

− µm[ωm1 + ωm2 − y −md1 −md2]

− µx[√

yδ − xd1 − xd2]

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where I am ignoring the xd1 ≥ 0, xd2 ≥ 0, y ≥ 0 constraints. We will seethat the solution will satisfy these. The system of first order conditionswrites

1 + µ∗m = 0

λ∗ + µ∗m = 0

1

2

1√x∗d1

+ µ∗x = 0

λ∗φ

2

1√x∗d2

+ µ∗x = 0

µ∗m − µ∗x1

2√y∗δ

= 0.

From these immediately µ∗m = −1 and λ∗ = 1 so that the inequality con-straint binds in optimum. The third and the fourth foc with λ∗ = 1 canbe solved for x∗d1φ

2 = x∗d2. The constraint associated with µx then gives√y∗ =

√δx∗d2(1 + 1

φ2).

Taking the fourth and fifth foc along with√y∗ =

√δx∗d2(1 + 1

φ2) and

λ∗ = 1 gives a system of two equations in two unknowns x∗d2, µ∗x. Solvingthe system and noting that we already know x∗d1φ

2 = x∗d2 as well as y∗ =δ(x∗d2)2(1 + 1

φ2)2 gives

x∗d1 =

[1

22δ(1 + φ2)

] 23

x∗d2 =

[φ3

22δ(1 + φ2)

] 23

µ∗x =

[(1 + φ2)δ

2

] 13

y∗ =

[1 + φ2

24√δ

] 23

.

Notice that all these are independent of u2 and hence have to be part of anyPareto efficient allocation. For given u2, m∗d1 and m∗d2, the only remainingvariables, can be computed from

m∗d2 = u2 − φ√x∗d2

m∗d1 = ωm1 + ωm2 − y∗ −m∗d2 = ωm1 + ωm2 − y∗ − u2 + φ√x∗d2.

These expressions explicitly show that m∗d1 and m∗d2 change one for one withu2. Denote them by m∗d1(u2) and m∗d2(u2). To summarize,

(x∗d1, x∗d2, y

∗,m∗d1(u2),m∗d2(u2))

for any u2 ∈ R constitutes a Pareto efficient allocation.

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solution: 2. Consumer i ∈ {1, 2} chooses (xdi,mdi) taking p and ωmi+θiπas given. The consumer’s problem is

maxxdi≥0,mdi∈R

mdi + φi−1√xdi s.t. mdi + pxdi ≤ ωmi + θiπ

The two first order conditions are (we are skipping some steps here, namelythose that take into account the xdi ≥ 0 constraint, we will see that x∗di willsatisfy x∗di ≥ 0)

1− λ∗i = 0

φi−1

2√x∗di− λ∗i p = 0

so that

xd1(p, ωm1, π) =1

4p2

xd2(p, ωm2, π) =φ2

4p2.

Because λ∗i = 1 the consumer’s budget constraint has to bind in optimumand hence

md1(p, ωm1, π) = ωm1 + θ1π − pxd1(p, ωm1, π) = ωm1 + θ1π −1

4p

md2(p, ωm2, π) = ωm2 + θ2π − pxd2(p, ωm2, π) = ωm2 + θ2π −φ2

4p.

solution: 3. Identical to question 4. Hence

xs(p) =p

2δms(p) =

p2

4δπ(p) =

p2

4δ

solution: 4. The easiest way to find p∗ is to clear the x market. Fromabove this means

xs(p∗) =

p∗

2δ= xd1(p∗, ωm1, π) + xd2(p∗, ωm2, π) =

φ2

4(p∗)2+

1

4(p∗)2

which implies

p∗ =

[δ(1 + φ2)

2

] 13

.

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Substituting p∗ into the expressions above gives

xd1(p∗, ωm1, π(p∗)) =1

4(p∗)2=

[1

24δ2(1 + φ2)2

] 13

= x∗d1

xd2(p∗, ωm2, π(p∗)) =φ2

4(p∗)2=

[φ6

24δ2(1 + φ2)2

] 13

= x∗d1

md1(p∗, ωm1, π(p∗)) = ωm1 + θ1π(p∗)− 1

4p∗= ωm1 +

[1

25δ(1+φ2)

] 13[θ1

1+φ2

2 − 1]

= m∗d1

md2(p∗, ωm2, π(p∗)) = ωm2 + θ2π(p∗)− φ2

4p∗= ωm2 +

[1

25δ(1+φ2)

] 13[θ2

1+φ2

2 − φ2]

= m∗d2

xs(p∗) =

p∗

2δ=

[1 + φ2

24δ

] 13

= x∗s

ms(p∗) =

(p∗)2

4δ=

[1 + φ2

24√δ

] 23

= m∗s

π(p∗) =(p∗)2

4δ=

[1 + φ2

24√δ

] 23

= π∗.

It is immediate that all markets clear. It is also immediate that the equilib-rium is Pareto efficient.

solution: 5. The intuition is similar as in question 4. The added insightcomes from the two consumers. If φ increases, consumer 2 values x more,its price p∗ increases, her demand for x, x∗d2, increases (which is not hard tocheck), and the supply side of the economy does better since all x∗s, m

∗s and

π∗ increase. Since x is more expensive consumer 1 reduces her demand andx∗d1 decreases.

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problem set 10, solution.pdf

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